Plane periodic patterns have been used as decorations all over the world for millennia. Apart from being used in the design of floor tiles, wallpapers, textiles, and other articles, plane periodic patterns also occur in nature as cross sections of crystal lattices. As such, crystallographers have studied them since the 19th century. It was then that scientists discovered that all the plane periodic patterns fall into just 17 categories. Mathematicians consider two patterns in the same category to be “the same”, even if the patterns look somewhat different to a layperson. This begs the question: what criteria are used to determine whether any two given patterns are in the same category, or, in other words, “the same”? The mathematically correct answer is that two patterns are “the same” if their groups of symmetry are the same. A pattern's group of symmetry is the set of all the moves (such as rotations, reflections, translations, or glide reflections) one can perform on a pattern without changing it. This definition of “the same” with regard to plane periodic patterns will be used throughout this document.
The “17 wallpaper patterns,” as they are commonly known, have been the subject of rigorous mathematical research, of which the most recent and complete exposition can be found in (Conway, Burgiel, & Goodman-Strauss, The Symmetries of Things, 2008, pp. 15-49). This work describes one of the standard notations for the patterns, known as the “Conway notation” and shows how this notation holds the key to the proof of why there are only 17 possible patterns. Conway et al. go further to show the relationships between the various patterns and the different ways they can be colored. They put these findings in a mathematical framework using group theory, which is an extremely important and powerful tool in the study of mathematics.
Given both the visual and mathematical appeal of planar patterns, they have been a popular subject of study for students of a variety of levels. At the college level, students are taught to identify each of the 17 patterns in textbooks such as (Gallian, 2009, pp. 467-478). For students wishing to create an electronic version of any of the 17 patterns based on a single motif, there are a number of free software packages available on the Internet. These include Escher Web Sketch (Nicolas Schoeni, Hardake, & Chapuis) and Java Kali (Amenta & Phillips, 1996). Both of these programs can create any of the 17 patterns in seconds, making them effective tools for demonstration. However, the speed and the method with which the patterns are drawn on the screen make it difficult for most users to learn about the structure and identification of the patterns from these programs. All the elements of the tiling array are drawn on the screen simultaneously, so that the user has a hard time grasping the symmetries (i.e. rotations, translations, glide reflections, or reflections) that generate the entire pattern.
There are infinitely many ways to create tessellations, and much work has been done in the area of designing tessellating articles. Some examples can be seen in U.S. Pat. No. 909,603 (1909) to Janin, U.S. Pat. No. 4,133,152 (1979) to Penrose, U.S. Pat. No. 4,620,998 (1986) to Lalvani, U.S. Pat. No. 5,619,830 (1997) to Osborn, and U.S. Pat. No. 6,309,716 (2001) to Fisher. The above patents disclose sets of tiles that can be assembled into either periodic or non-periodic planar patterns. However, none can construct the complete set of plane periodic patterns, nor teach a means to make this possible.
U.S. Pat. No. 5,368,301 (1994) issued to Mitchell discloses a puzzle composed of isosceles right triangle pieces that are similar in shape to some of the pieces in the present invention, as well as equilateral triangles. However, it is not possible to construct the complete set of all 17 repeating plane patterns with this set, even if one were to work with a multitude of identical tiles.
U.S. Pat. No. 7,833,077 (2010) due to Simmons, Jr. discloses interlocking blocks that can be assembled in a variety of ways. Their cross sections are somewhat similar to the tiles in the present invention, but they cannot be assembled into all of the 17 repeating planar patterns.
U.S. Pat. No. 3,633,286 (1972) to Maurer and U.S. Pat. No. 5,203,706 (1993) to Zamir disclose interlocking tiles with a stencil portion. Both of these inventions can be used to draw a very limited number of repeating planar patterns due to the shape of the tiles (rectangular) and the placement of the interlocking elements.
There are even fewer teaching resources that can be extended to three dimensions. The Platonic solids are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids: the cube, dodecahedron, icosahedron, octahedron, and tetrahedron. This fact was proved by Euclid in the last proposition of the Elements. (see Weisstein, Eric W. “Platonic Solid.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/PlatonicSolid.html visited 13 Nov. 2014). The Platonic solids, when projected on a sphere, tessellate it into regular spherical polygons. Thus, they can be viewed as regular tessellations of the sphere (http://en.wikipedia.org/wiki/Platonic_solid). There is a need for a teaching tool that can demonstrate the construction of all 17 symmetry types of plane repeating patterns. There is a need to extend this teaching set to three dimensions and be capable of creating three-dimensional solid elements, such as all five Platonic solids, and to extend the symmetric patterns on spherical surfaces (see Conway p. 51). There is a need for a teaching tool that can go beyond this and also create other well known solid types such as the Archimedean solids. The Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides of the polygons the same length. There are 13 Archimedean solids. (see Weisstein, Eric W. “Archimedean Solid.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedeanSolid.html, visited 13 Nov. 2014).